Restrictions in Multiphenicsx

General
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1

Hello,

I’m currently working on multiphenicsx and i have questions regarding the use of restrictions and defining boundary conditions and how boundary conditions are applied in blocked formulation.

The Setup is slightly modified as described in this post.

Quick Summary of the Setup:
Im currently working on a simple example of a 3 Phase domain consists of YSZ-Phase, Nickel-Phase and Gas-Phase, (see the image). The Ionic Potential, Electronic Potential and also Partial Pressure of Water and Hydrogen gas are governed by the 4 electrochemical equations respectively and there is a coupled term j at the triple-phase-boundary between the 3 Phases. I have restricted the degrees of freedom (DOFs) to be active only within the respective region for each electrochemical equations.

\nabla \cdot (\sigma_{\text{io}} \nabla V_{\text{io}}) = -\ j \quad \text{in} \, \Omega_{\text{io}}
\nabla \cdot (\sigma_{\text{el}} \nabla V_{\text{el}}) = 2 \ j \quad \text{in} \, \Omega_{\text{el}}
\nabla \cdot \left( \frac{D_{H_2O}}{R T} \nabla P_{H_2O} \right) = \frac{j}{2F} \quad \text{in} \, \Omega_{\text{pore}}
\nabla \cdot \left( \frac{D_{H_2}}{R T} \nabla P_{H_2} \right) = -\frac{j}{2F} \quad \text{in} \, \Omega_{\text{pore}}

image
image941×480 72.7 KB

Questions:

  1. Behavior of DOFs outside of the restricted subdomains:
    When I apply restrictions to the \Omega_{\text{io}}, are the DOFs in other domains are effectively set to zero or ignored within this restricted system?
    Thus, are Dirichlet boundary conditions applied for the potential at other phases or can the interface between the phases be regarded as external boundaries where Neumann boundary conditions can applied?
    How would it affect the solution in \Omega_{\text{el}}?

  2. Application of Neumann Boundary Conditions at the interface 12:
    Suppose i want to impose a non-homogeneous Neumann boundary condition on the boundary 12 of \Omega_{\text{io}}. This would add an additional term to the weak formulation -\int_{\Omega_{\text{N}}} \sigma_{\text{io}} \nabla V_{\text{io}}\cdot nds and would this only affect the block corresponding to \Omega_{\text{io}}?
    Do I need to use ds oder dS?

If a MWE is required, please feel free to tell me.
I appreciate your help, thank you for your time!

2
  1. While assembling the system, DOFs outside of the restriction are effectively ignored.
  2. You’ll need to use dS, because the interface is not on the boundary.